Lavaei Department of Electrical Engineering Columbia University Joint work with Somayeh Sojoudi Convexification of Optimal Power Flow Problem by Means of Phase Shifters Power Networks ID: 400677
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Javad LavaeiDepartment of Electrical EngineeringColumbia UniversityJoint work with Somayeh Sojoudi
Convexification of Optimal Power Flow Problem by Means of Phase ShiftersSlide2
Power Networks
Optimizations: Optimal power flow (OPF) Security-constrained OPF State estimation Network reconfiguration Unit commitment
Dynamic energy management
Issue of non-convexity:
Discrete parameters
Nonlinearity in continuous variables
Transition from traditional grid to smart grid: More variables (10X) Time constraints (100X)
Javad Lavaei, Columbia University
2Slide3
Broad Interest in Optimal Power FlowJavad Lavaei, Columbia University3 OPF-based problems solved on different time scales:
Electricity marketReal-time operationSecurity assessmentTransmission planning Existing methods based on linearization or local search
Question:
How to find the best solution using a scalable robust algorithm?
Huge literature since 1962 by power, OR and Econ people
Slide4
Summary of ResultsJavad Lavaei, Columbia University4 A sufficient condition to globally solve OPF:
Numerous randomly generated systems IEEE systems with 14, 30, 57, 118, 300 buses European grid Various theories: It holds widely in practice
Project 1:
How to solve a given OPF in polynomial time?
(joint work with Steven Low)
Distribution networks are fine (under certain assumptions).
Every transmission network can be turned into a good one
(under
assumptions
).
Project 2:
Find network topologies over which optimization is easy?
(joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)
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Summary of ResultsJavad Lavaei, Columbia University5Project 3:
How to design a distributed algorithm for solving OPF? (joint work with Stephen Boyd, Eric Chu and Matt Kranning) A practical (infinitely) parallelizable algorithm It solves 10,000-bus OPF in 0.85 seconds on a single core machine.
Project 4:
How to do optimization for mesh networks?
(joint work with
Ramtin
Madani and Somayeh Sojoudi)
Developed a penalization technique
Verified its performance on IEEE systems with 7000 cost functions
Focus of this talk:
Revisit Project 2 and remove its assumptionsSlide6
Geometric Intuition: Two-Generator Network
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Optimal Power FlowCost OperationFlowBalance
SDP relaxation:
Remove the rank constraint.
Exactness of relaxation:
We study it thru a geometric approach.
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Acyclic Three-Bus Networks
Assume
that the voltage magnitude is fixed at every bus.
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Geometric Interpretation
(+,+)
Pareto face:
Pareto face
Convex Pareto Front:
Injection region and its convex hull share the same front.
Javad Lavaei, Columbia University
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Two-Bus Network Two-bus network with power constraints:
P
1
P
2
P
1
P
2
P
1
P
2
P
1
P
2
P
1
P
2
P
1
P
2
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General Tree Network Assume that each flow-restricted region is already Pareto (monotonic curve):
PijPji
Ratio from 1 to 10:
Max angle from 45
o
to 80
o
Javad Lavaei, Columbia University
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Three-Bus Networks Issues: Coupling thru angles and voltage magnitudes
Variable voltage magnitude:
Javad Lavaei, Columbia University
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Decoupling Angles Phase shifter: An ideal transformer changing a phase Phase shifter kills the angles coupling.
PS
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Decoupling Voltage Magnitudes Define:
Boundary
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Injection & Flow Regions Voltage coupling introduces linear equations in a high-dimensional space.
Line (i,j):
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Main Result Current practice in power systems:Tight voltage magnitudes.Not too large angle differences.
Adding virtual phase shifters is often the only relaxation needed in practice.
Javad Lavaei, Columbia University
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Phase ShiftersJavad Lavaei, Columbia University17
Blue:
F
easible set (P
G1
,P
G2
)
Green:
Effect of phase shifter
Red:
Effect of convexification
Minimization over green = Minimization over green and red (even with box constraints)Slide18
Phase ShiftersSimulations: Zero duality gap for IEEE 30-bus system Guarantee zero duality gap for all possible load profiles? Theoretical side: Add 12 phase shifters Practical side: 2 phase shifters are enough
IEEE 118-bus system needs no phase shifters (power loss case)
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Phase shifters speed up the computation:Slide19
ConclusionsFocus: OPF with a 50-year historyGoal: Find a near-global solution efficiently
Main result: Virtual phase shifters make OPF easy under tight voltage magnitudes and not too loose angle differences. Future work: How to lessen the effect of virtual phase shifters?
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